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Kahun Papyrus (From Wikipedia)
The Kahun Papyrus (KP) is as an ancient Egyptian text discussing mathematical and medical topics. KP fragments were discovered by Flinders Petrie in 1889 and are kept at the University College London. Most of the texts were dated to ca 1825 BC, to the reign of Amenemhat III. One of its fragments, referred to as the Kahun Gynaecological Papyrus deals with gynecological illnesses and conditions.
The Kahun Papyrus (KP) is notable for an Egyptian fraction solution to an arithmetic progression problem. Ancient Egyptian fractions used an arithmetic notation represented by positive rational numbers, 2/n, or m/n rational numbers. The rational numbers were written as optimized, but not optimal, unit
fraction sums typically 5-terms 1/a + 1/b + 1/c + 1/d + 1/e or less, scaled by an LCM, and written in arithmetic operations that scholars confused as unique to the period. The Egyptian fraction notation was continuously used for over 3,000 years within modern-like arithmetic operations that had scholars had not parsed from Egyptian fractions methodologies until the 21st century AD. Revised forms of the notation began to die after 1454 AD with the rise and dominance of the algorithmic base 10 decimal arithmetic, approved by the Paris Academy in 1585 AD, and updated by Napier and others.
One fragment of the KP began with a traditional 2/n table, a Middle Kingdom scribe's method of defining a rational number as an exact unit fraction series. The KP 2/n table was abbreviated version of the Rhind Mathematical Papyrus (RMP) 2/n table and 51 rational numbers, 2/3 to 2/101. Considering the KP's arithmetic topics, arithmetic progressions waslikely the highest form of Egyptian arithmetic. The KP scribe defined a 10-term arithmetic progression summed to 100, with a difference (d) of 5/6. The KP arithmetic progression was discussed in two RMP problems.
To imply a generalized form of arithmetic progressions, the RMP scribe Ahmes listed two columns of data (published by Gillings in 1972). Ahmes's thinking is shown in Gillings' column 11 by multiplying 5/12 times 9, a fact that was needed to find the largest term of the RMP progression. Ahmes then added 10 and wrote out the correct largest term of the arithmetic progression, and subtracted 5/6, nine times. Gillings found the remaining terms of the progressions by using the KP's method. To understand the KP method, readers must make arithmetic calculations as the Middle Kingdom scribes wrote down in their three problems, double and triple checking your work with several tools.
Gillings' 1972 analysis of both RMP versions of Middle Kingdom arithmetic progression failed to parse the method in a manner that was comparable, in every respect, to the KP method. For example, Gillings had noticed similar problems in the RMP (RMP 40, 64) yet Gillings muddled three pages of his analysis, thereby reaching no definitive conclusions on this topic.
In 1987, Egyptologist Gay Robins, and Charles Shute, wrote on the Rhind Mathematical Papyrus(RMP). Five years later, Egyptologist John Legon wrote on the KP, and closely related arithmetic proportions in the RMP. The KP and RMP report scribal uses the same method to find the largest term of closely related arithmetic progressions. The method: take 1/2 of the difference, 1/2 of 5/6 (5/12 in the KP) times the number of differences (nine times 5/12 = 15/4 in the KP) plus the sum of the A.P progression (100 in the KP) divided by the number of terms (10 , meaning 100/10 = 10 in the KP). Finally add column 11's result, 3 3/4, to 10, and the largest term, 13 3/4.
To repeat, add column 11, 5/12 times 9, or 45/12, or 3 3/4, 3 2/3 1/12 in Egyptian fractions to 10 in column 12 beginning with the largest term 13 2/3 1/12. The scribe subtracted 5/6 nine times, creating the remaining terms of the progression.
Robins-Shute confused an aspect of the problem by omitting the sum divided by the number of terms parameter in the RMP. An algebraic statement could have been created by Robins-Shute from matched pairs that added to 20, five pairs summing to 100, as potentially related to RMP 40.
The KP method found the largest term, and used other facts that have been reported in RMP 64, and RMP 40, by John Legon in 1992. Scholars, at other times, have attempted to parse Rhind Mathematical Papyrus 40, a problem that asks 100 loaves of bread to be shared between five men by finding the smallest term of an arithmetic progression.
A confirmation of the Kahun method is reported by RMP 64, and RMP 40. In RMP 64 Ahmes asked 10 men to share 10 hekats of barley, with a differential of 1/8, by using an arithmetical progression? Robins and Shute reported, "the scribe knew the rule that, to find the largest term of the arithmetical progression, he must add half the difference to the average number of terms as many times as there are common differences, that is, one less than the number of terms" (note that Robins-Shutre omitted the sum divided by the number of terms), as noted by:
1. number of terms: 10
2. arithmetical progression difference: 1/8
3. arithmetic progression sum: 10
The scribe used the following facts to find the largest term.
1. one-half of differences, 1/16, times number of terms minus one, 9,
1/16 times 9 = 9/16
2. The computed parameter(1), was found by 10, the sum, divided by 10, the number of terms. It was inserted by Robins-Shute, but had not been high-lighted, citing 1 + 1/2 + 1/16, or 1 9/16, the largest term. The remaining nine terms were found by subtracting 1/8 nine times to obtain the remaining barley shares.
That is, the KP scribe used formula 1.0:
(1/2)d(n-1) + S/n = Xn (formula 1.0)
with,
d = differential, n = number of terms in the series, S = sum of the series, Xn = largest term in the series allowed three(of the four) parameters: d, n, S and Xn, to algebraically find the fourth parameter. When n was odd, x (n/2) = S/n, and x 1 + xn = x2 + x(n -1) = x3 + x(n -2) = ... = x(n/2) = S/n, a paired data set that Carl Friedrich Gauss implemented as a grammar school student solving the n = even case. Ahmes and Gauss found the sum of 1 to 100, using d = 1, by following the same rule. Both reached 5050 based on 50 pairs of 101 (1 + 101 = 2 + 99 = 3 + 98 = ...) .
In summary, KP and RMP scribes used an identical method to calculate the largest term in arithmetic progressions, as well as finding any one of the four variables in formula 1.0 when three variables (d, n, S and xn) were known. Note that formula 1.0 did not rely on the rational number differences (d) being generally converted to Egyptian fraction series. Despite agreement on the larger questions, minor questions remain open for scholars. For example, what were the beginning, and intermediate arithmetic steps, of the arithmetic progression formula, particularly scribal subtraction and division operations? Asking the question in other terms, why were vulgar fractions present in beginning and intermediate calculations in the KP, and the RMP? Other common calculations were used in the KP and the RMP. The KP scribe created Egyptian fractions as final statements based on Egyptian fraction arithmetic, writing beginning, and intermediate vulgar fractions in identical ways. Two exceptions were the calculation of arithmetic progressions, and weights and measures calculations, i.e. hekat (volume) units.
- 1
- Richard Gillings, "Mathematics in the Time of the Pharaohs", pages 176-180, MIT Press, Cambridge, 1972
- 2
- John Legon, "A Kahun Papyrus Fragment", pages 21-24, Discussions in Egyptology 24, 1992.
- 3
- Luca Miatello, "The difference 5 1/2 in a problem of rations from the Rhind mathematical papyrus", Historia Mathematica, vol 34, issue 4, pages 277-284, Nov. 2008.
- 4
- Gay Robins and Charles Shute, "The Rhind Mathematical Papyrus", pages 41-43, British Museum Press, Dover Reprint, 1987.
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